Recently, neural networks have demonstrated remarkable capabilities in mapping two arbitrary sets to two linearly separable sets. The prospect of achieving this with randomly initialized neural networks is particularly appealing due to the computational efficiency compared to fully trained networks. This paper contributes by establishing that, given sufficient width, a randomly initialized one-layer neural network can, with high probability, transform two sets into two linearly separable sets without any training. Moreover, we furnish precise bounds on the necessary width of the neural network for this phenomenon to occur. Our initial bound exhibits exponential dependence on the input dimension while maintaining polynomial dependence on all other parameters. In contrast, our second bound is independent of input dimension, effectively surmounting the curse of dimensionality. The main tools used in our proof heavily relies on a fusion of geometric principles and concentration of random matrices.
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A near-field wideband communication system is studied, wherein a base station (BS) employs an extremely large-scale antenna array (ELAA) to serve multiple users situated within its near-field region. To facilitate the near-field beamfocusing and mitigate the wideband beam split, true-time delayer (TTD)-based hybrid beamforming architectures are employed at the BS. Apart from the fully-connected TTD-based architecture, a new sub-connected TTD-based architecture is proposed for enhancing energy efficiency. Three wideband beamfocusing optimization approaches are proposed to maximize spectral efficiency for both architectures. 1) Fully-digital approximation (FDA) approach: In this approach, the TTD-based hybrid beamformers are optimized to approximate the optimal fully-digital beamformers using block coordinate descent. 2) Penalty-based FDA approach: In this approach, the penalty method is leveraged in the FDA approach to guarantee the convergence to a stationary point of the spectral maximization problem. 3) Heuristic two-stage (HTS) approach: In this approach, the closed-form TTD-based analog beamformers are first designed based on the outcomes of near-field beam training and the piecewise-near-field approximation. Subsequently, the low-dimensional digital beamformer is optimized using knowledge of the low-dimensional equivalent channels, resulting in reduced computational complexity and channel estimation complexity. Our numerical results unveil that 1) the proposed approaches effectively eliminate the near-field beam split effect, and 2) compared to the fully-connected architecture, the proposed sub-connected architecture exhibits higher energy efficiency and imposes fewer hardware limitations on TTDs and system bandwidth.
Most conventional crowd counting methods utilize a fully-supervised learning framework to establish a mapping between scene images and crowd density maps. They usually rely on a large quantity of costly and time-intensive pixel-level annotations for training supervision. One way to mitigate the intensive labeling effort and improve counting accuracy is to leverage large amounts of unlabeled images. This is attributed to the inherent self-structural information and rank consistency within a single image, offering additional qualitative relation supervision during training. Contrary to earlier methods that utilized the rank relations at the original image level, we explore such rank-consistency relation within the latent feature spaces. This approach enables the incorporation of numerous pyramid partial orders, strengthening the model representation capability. A notable advantage is that it can also increase the utilization ratio of unlabeled samples. Specifically, we propose a Deep Rank-consistEnt pyrAmid Model (DREAM), which makes full use of rank consistency across coarse-to-fine pyramid features in latent spaces for enhanced crowd counting with massive unlabeled images. In addition, we have collected a new unlabeled crowd counting dataset, FUDAN-UCC, comprising 4,000 images for training purposes. Extensive experiments on four benchmark datasets, namely UCF-QNRF, ShanghaiTech PartA and PartB, and UCF-CC-50, show the effectiveness of our method compared with previous semi-supervised methods. The codes are available at //github.com/bridgeqiqi/DREAM.
Popular methods in compressed sensing (CS) are dependent on deep learning (DL), where large amounts of data are used to train non-linear reconstruction models. However, ensuring generalisability over and access to multiple datasets is challenging to realise for real-world applications. To address these concerns, this paper proposes a single image, self-supervised (SS) CS-MRI framework that enables a joint deep and sparse regularisation of CS artefacts. The approach effectively dampens structured CS artefacts, which can be difficult to remove assuming sparse reconstruction, or relying solely on the inductive biases of CNN to produce noise-free images. Image quality is thereby improved compared to either approach alone. Metrics are evaluated using Cartesian 1D masks on a brain and knee dataset, with PSNR improving by 2-4dB on average.
The common spatial pattern analysis (CSP) is a widely used signal processing technique in brain-computer interface (BCI) systems to increase the signal-to-noise ratio in electroencephalogram (EEG) recordings. Despite its popularity, the CSP's performance is often hindered by the nonstationarity and artifacts in EEG signals. The minmax CSP improves the robustness of the CSP by using data-driven covariance matrices to accommodate the uncertainties. We show that by utilizing the optimality conditions, the minmax CSP can be recast as an eigenvector-dependent nonlinear eigenvalue problem (NEPv). We introduce a self-consistent field (SCF) iteration with line search that solves the NEPv of the minmax CSP. Local quadratic convergence of the SCF for solving the NEPv is illustrated using synthetic datasets. More importantly, experiments with real-world EEG datasets show the improved motor imagery classification rates and shorter running time of the proposed SCF-based solver compared to the existing algorithm for the minmax CSP.
Neural operators extend the capabilities of traditional neural networks by allowing them to handle mappings between function spaces for the purpose of solving partial differential equations (PDEs). One of the most notable methods is the Fourier Neural Operator (FNO), which is inspired by Green's function method and approximate operator kernel directly in the frequency domain. In this work, we focus on predicting multiscale dynamic spaces, which is equivalent to solving multiscale PDEs. Multiscale PDEs are characterized by rapid coefficient changes and solution space oscillations, which are crucial for modeling atmospheric convection and ocean circulation. To solve this problem, models should have the ability to capture rapid changes and process them at various scales. However, the FNO only approximates kernels in the low-frequency domain, which is insufficient when solving multiscale PDEs. To address this challenge, we propose a novel hierarchical neural operator that integrates improved Fourier layers with attention mechanisms, aiming to capture all details and handle them at various scales. These mechanisms complement each other in the frequency domain and encourage the model to solve multiscale problems. We perform experiments on dynamic spaces governed by forward and reverse problems of multiscale elliptic equations, Navier-Stokes equations and some other physical scenarios, and reach superior performance in existing PDE benchmarks, especially equations characterized by rapid coefficient variations.
Recent work in data-driven modeling has demonstrated that a weak formulation of model equations enhances the noise robustness of a wide range of computational methods. In this paper, we demonstrate the power of the weak form to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a recently developed data-driven reduced order modeling technique. We introduce a weak form-based version WLaSDI (Weak-form Latent Space Dynamics Identification). WLaSDI first compresses data, then projects onto the test functions and learns the local latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. With WLaSDI, the local latent space is obtained using weak-form equation learning techniques. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and accurate simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common benchmark examples including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in the case of 1D inviscid Burgers' simulations with the addition of up to 100% Gaussian white noise, the relative error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for LaSDI. Similarly, for radial advection simulations, the relative errors stay below 15% for WLaSDI, in stark contrast to the potential errors of up to 10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X speedup compared to the corresponding full order model. Python code to reproduce the results in this work is available at (//github.com/MathBioCU/PyWSINDy_ODE) and (//github.com/MathBioCU/PyWLaSDI).
Social relations are often used to improve recommendation quality when user-item interaction data is sparse in recommender systems. Most existing social recommendation models exploit pairwise relations to mine potential user preferences. However, real-life interactions among users are very complicated and user relations can be high-order. Hypergraph provides a natural way to model complex high-order relations, while its potentials for improving social recommendation are under-explored. In this paper, we fill this gap and propose a multi-channel hypergraph convolutional network to enhance social recommendation by leveraging high-order user relations. Technically, each channel in the network encodes a hypergraph that depicts a common high-order user relation pattern via hypergraph convolution. By aggregating the embeddings learned through multiple channels, we obtain comprehensive user representations to generate recommendation results. However, the aggregation operation might also obscure the inherent characteristics of different types of high-order connectivity information. To compensate for the aggregating loss, we innovatively integrate self-supervised learning into the training of the hypergraph convolutional network to regain the connectivity information with hierarchical mutual information maximization. The experimental results on multiple real-world datasets show that the proposed model outperforms the SOTA methods, and the ablation study verifies the effectiveness of the multi-channel setting and the self-supervised task. The implementation of our model is available via //github.com/Coder-Yu/RecQ.
Learning latent representations of nodes in graphs is an important and ubiquitous task with widespread applications such as link prediction, node classification, and graph visualization. Previous methods on graph representation learning mainly focus on static graphs, however, many real-world graphs are dynamic and evolve over time. In this paper, we present Dynamic Self-Attention Network (DySAT), a novel neural architecture that operates on dynamic graphs and learns node representations that capture both structural properties and temporal evolutionary patterns. Specifically, DySAT computes node representations by jointly employing self-attention layers along two dimensions: structural neighborhood and temporal dynamics. We conduct link prediction experiments on two classes of graphs: communication networks and bipartite rating networks. Our experimental results show that DySAT has a significant performance gain over several different state-of-the-art graph embedding baselines.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.
The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely. Experiments on two machine translation tasks show these models to be superior in quality while being more parallelizable and requiring significantly less time to train. Our model achieves 28.4 BLEU on the WMT 2014 English-to-German translation task, improving over the existing best results, including ensembles by over 2 BLEU. On the WMT 2014 English-to-French translation task, our model establishes a new single-model state-of-the-art BLEU score of 41.8 after training for 3.5 days on eight GPUs, a small fraction of the training costs of the best models from the literature. We show that the Transformer generalizes well to other tasks by applying it successfully to English constituency parsing both with large and limited training data.
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